As I’ve been trying to point out – and as others, notably Ben Bernanke, have also tried to point out – such monetary wisdom as we possess starts with Knut Wicksell’s concept of the natural interest rate. Try to keep rates too low, and inflation accelerates; try to keep them too high, and inflation decelerates and heads toward deflation.So, I was thinking, what happens if we write that down and work it out?
To keep it simple, we'll just deal with a deterministic world. It's more or less New Keynesian, but a little different. To start, we have the standard Euler equation, which prices a one-period nominal bond - after taking logs and linearizing:
(1) R(t) = r* + ag(t+1) + i(t+1),
where R(t) is the nominal interest rate, r* is the subjective discount rate, a is the coefficient of relative risk aversion (assumed constant), g(t+1) is the growth rate in consumption between period t and period t+1, and i(t+1) is the inflation rate, between period t and period t+1. Similarly, the real interest rate is given by
(2) r(t) = r* + ag(t+1).
Assume there is no investment, and all output is consumed.
To capture Krugman's concept of Wicksellian inflation dynamics, first let r* + ag* denote the Wicksellian natural rate of interest, where g* is the economy's long-run growth rate. Krugman says that inflation goes up when the the real interest rate is low relative to the natural rate, and inflation goes down when the opposite holds. So, write this as a linear relationship,
(3) i(t+1) - i(t) = -b[r(t) - r* - ag*],
where b > 0. Then, from (2) and (3),
(4) i(t) = ba[g(t+1)-g*] + i(t+1),
which is basically a Phillips curve - given anticipated inflation, inflation is high if the growth rate of output is high.
Then, substitute for g(t+1) in equation (1), using (4), and write
(5) i(t+1) = -[b/(1-b)][R(t) - r* - ag*] + [1/1-b]i(t).
So this is easy now, as to determine an equilibrium we just need to solve the difference equation (5) for the sequence of inflation rates, given some path for R(t), or some policy rule for R(t), determined by the central bank.
First, suppose that R(t) = R, a constant. Then, from (5), the unique steady state is
(6) i = R - r* - ag*.
That's just the long-run Fisher relation - the inflation rate is the nominal interest rate minus the natural real rate of interest. But what about other equilibria? If 0 < b < 1, or b > 2, then in fact the steady state given by (6) is the only equilibrium. If 1 < b < 2 then there are many equilibria which all converge to the steady state.
Next, suppose that R(t) = R1, for t = 0, 1, 2, ..., T-1, and R(t) = R2, for t = T, T+1, T+2,..., where R2 > R1. This is an experiment in which the nominal interest rate goes up, once and for all, at time T, and this change in monetary policy is perfectly anticipated. In the case where 0 < b < 1, there is a unique equilibrium that looks like this:
So, inflation increases prior to the nominal interest increase, and achieves the Fisherian steady state in period T, and the growth rate in output and the real interest rate are low and falling before the nominal interest rate increase occurs.
We can look at the other cases, in which b > 1, and the dynamics will be more complicated. Indeed, we get multiple equilibria in the case 1 < b < 2. But, in all of these cases, a higher nominal interest rate implies convergence to the Fisherian steady state with a higher inflation rate. Increasing the nominal interest rate serves to increase the inflation rate. Keeping the nominal interest rate at zero serves only to keep the inflation rate low, in spite of the fact that this model has Wicksellian dynamics and a Phillips curve.
I'm not endorsing this model - just showing you its implications. And those implications certainly don't conform to "try to keep rates too low, and inflation accelerates; try to keep them too high, and inflation decelerates and heads toward deflation," as Krugman says. The Wicksellian process is built into the model, just as Krugman describes it, but the model has neo-Fisherian properties.